Question

a right triangle has a hypotenuse of length $10\sqrt{2}$ and an angle of 45 degrees, with a side opposite this angle with a length of 10. a second right triangle also has an angle of 45 degrees, with a side opposite this angle with a length of 14. determine the length of the hypotenuse in the second triangle. (1 point)

○ the hypotenuse of the second triangle has a length of 14
○ the hypotenuse of the second triangle has a length of $7\sqrt{2}$
○ the hypotenuse of the second triangle has a length of $14\sqrt{2}$
○ the hypotenuse of the second triangle has a length of 7

Explanation:

Step1: Recall sine definition

For a right triangle, $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

Step2: Solve for hypotenuse

Rearrange formula: $\text{hypotenuse} = \frac{\text{opposite}}{\sin(\theta)}$
Substitute values: $\text{opposite}=14$, $\theta=45^\circ$, $\sin(45^\circ)=\frac{\sqrt{2}}{2}$
$\text{hypotenuse} = \frac{14}{\frac{\sqrt{2}}{2}} = 14 \times \frac{2}{\sqrt{2}} = 14\sqrt{2}$

Answer:

The hypotenuse of the second triangle has a length of $14\sqrt{2}$